Abstract
The Alternating Gradient
Synchrotron (AGS) employs two partial helical snakes[1]
to preserve the polarization of the proton beam during acceleration. In order
to compensate for the adverse effect of the partial helical snakes on the beam
optics in the AGS during acceleration of the beam, we introduced eight
“compensation” quadrupoles in straight sections of
the AGS at the proximity of the partial snakes. At injection energies, the
strength of these eight quads is set at a high value but is ramped down to zero
when the effect of the snakes diminishes due to the increase of beam’s rigidity.
Four of the eight compensation quadrupoles had to be
placed in very short straight sections therefore had to be “thin” with a length
of ~30 cm. In this paper we will discus: a) The
mechanical and magnetic specifications of the “thin” quadrupoles.
b) the method to minimize the strength of the dodecapole harmonic. c) The method to optimize the
thickness of the laminations that the magnet iron is
made. d) mechanical tolerances of the magnet e) comparison
of the measured and calculated magnetic multipoles of
the quadrupole.
1
Mechanical specifications
Figure
1 shows an isometric
view of the iron core of the “thin” quadrupole.
Figure
1: Isometric view of the “thin” quadrupole. The shape
of the pole face is also shown.
The
radius of the quadrupole’s apperture
and the length, width, and height of the pole piece appears in Table 1. Figure
2 shows one of the coils placed around one of the pole pieces.
Table 1: Mechanical dimensions of
the “thin” quad.
|
R
cm |
Length
[cm] |
Width
[cm] |
Height
[cm] |
Lcoil cm |
|
8.3 |
10.3 |
10.2 |
14.0 |
18.0 |
Each
of the four coils is square and the length of the outer side is 18 cm. The coil
is made of four layers with thirteen turns per layer. The water cooled
conductor allows a maximum current of 350 A in the conductor which is of square
cross section of 8 x 8 mm2 and a circular inner diameter of 4.3 mm
for water cooling.
Figure
2: An isometric view of the magnet’s coil around one of the quadrupole’s
pole.
2 MAGNETIC specifications
The
magnetic specification of the “thin” quadrupole
are: a) The required integrated gradient ∫Gdl of the “thin” quadrupole must have the
value 0.76 [T] or higher at its maximum current. b) The strength of the first
allowed multipole (12-pole) must be below an upper
limit. This limit was determined by the strength of the feed-down sextupole that is introduced due to a possible transverse
misplacement of the quadrupole. c) An upper limit of
the possible mechanical tolerances was also established, by setting an upper limit
on the sextupole caused by the various possible mechanical
misalignments. d) Minimize the adverse effects of the eddy currents in the iron
core, on the magnetic field.
3 MAGNETIC MODELLING
In
order to satisfy the magnetic requirements of the “thin” quadrupole,
we performed 2D and 3D magnetic modelling[2] of the
thin quadrupole.
3.1 2D Modelling
The
2D magnetic modelling was performed to establish: a) the contour of the iron
pole-face that minimizes the dodecapole multipole. The contour of the pole faces is shown in Fig. 1
and also in Fig. 2. b) to estimate homic losses which are
generated in the magnet coils because of the eddy currents. The ohmic losses in the coils due to the eddy currents generated
during the ramp-down of the magnet, were calculated to be 0.8% of the ohmic losses due to the main current which powers the
magnet, c) to provide reasonable amount of iron so that the iron in the return leg is not saturated.
3.2 3D Modelling and results
The
purpose of the 3D magnetic field calculations is discussed in the following
sections.
3.2.1 Minimization of the dodecapole
multipole
Static
calculations were performed on a 3D model in order to determine the amount of
“chamfer” of the pole pieces, at the entrance and exit of the magnet, which
minimizes the strength of the integrated dodecapole.
This “chamf” of the pole pieces which is shown in
Figure 1 and 2 was determined to be 22.8o and started 1.27 cm from
the edge of each pole piece. The combination of the pole face contour as
determined in the 2D calculations, and the “chamfering” of the pole pieces
determined in the 3D calculations reduced the integrated strength of the dodecapole field at a radius r=7 cm to 0.02% of the quadrupole field.
3.2.2 Transient Field Calculations
The
3D transient calculations were performed by using the ELEKTRA module of the
opera code[2] to help determine an upper limit in the
lamination thickness of the magnet iron. The required lamination thickness
should be such that it limits the ohmic losses, due
to eddy currents, below 10 [J] per acceleration-cycle, and also the maximum
field achieved by the quadrupole during the 200 msec ramping to be at least 99% of the static field
generated by the magnet when it is excited at the same current. For a given
lamination thickness, the coil current was ramped from 0 [A] to 350 [A] in 200 msec and, the gradient of the quadrupole
and the power dissipated in the laminations were calculated. We repeated the
calculations for a different lamination thickness but always keeping the length
of the magnet’s iron fixed at 10.3 [cm].
The magnetic permeability of the lamination material was non-linear and
similar to that of steel 1010. Fig. 3 shows the Gradiend/Icoil as a function of ramping time for
various thicknesses of the laminations. As seen in this figure for a lamination
thickness 0.595 [cm] the gradient of the quadrupole
is almost the same as that of the static field thus the effect of the eddy
currents is not significant.
Figure
3: The gradient of the “thin” quadrupole as a
function of time during which the current is ramped linearly from 0 to 350 [A].
Each plot corresponds to a different lamination thickness.
In
Fig. 4 ploted are the ohmic
losses in laminations, during ramping for a lamination thickness.
Figure
4: Power dissipated in the laminations as a function of time. Each plot
corresponds to a different lamination thickness.
The
lowest power loss occurs for the
lamination thickness 0.595 [cm] and it amounts to a total energy of 9 J,
dissipated in the iron of the magnet per acceleration cycle. It is noteworthy, as
shown by the negative slope of some of
the curves in Fig. 4, the reduction of the power dissipated in the iron as the
field in the iron increases. This can be explained by the “skin depth” d=(2/wm0ms)-1/2
increase. The increase of (d) is due to the decrease of
(m), as the magnetic field increases.
Thus if the value of the skin depth approaches the thickness of the laminations
the eddy currents which flow in opposite directions in the laminations cancel
each other, and the ohmic losses are reduced. This is
shown in Fig. 5 and 6, each shown an isometric view of the eddy current density
formed in the same lamination but at two different times. The eddy currents
shown at a letter time (Fig. 6) have greater skin depth and they partially
cancel each other.
3.2.3
Mechanical
tolerances
In
this study we used the full 3D model of the magnet because the symmetry of the
model is broken under a misalignment. The study showed that by displacing
laterally (azimuthally) or radially
(away from the quadrupole axis) one of the poles of
the magnet by ±0.25 [mm] the strength of the generated sextupole
multiple is well below the maximum permissible limit.
4 MECHANICAL DESIGN AND ASSEMBLY
4.1 Iron Core
The
“thin” quad magnet was required to be installed around an existing accelerator beam
pipe without breaking vacuum, thus the construction had to be modular, and
"self-fixtured" in order to maintain its
prescribed tight tolerances. So the basic engineering design approach was to
build two halves and then assemble them in
situ around the existing beam pipe.
Furthermore,
due to the very tight space allocated for its location in between two adjacent
large AGS (Alternating Gradient Synchrotron) ring magnets, we had no choice but
to split the “thin” quad magnet into quarter sections in order to allow the
assembly of the coil pack into the iron core, being very careful that the
symmetry and the tight assembly tolerances were preserved.
As indicated
in Fig.3, in order to avoid any significant eddy current effects, the
lamination thickness had to be less than 0.595 cm. (0.234 in). Hence, we decided to use a stock 0.25-in.
extra-low carbon steel plate, alloy 1005; both rough sides of the plate were
machined and surface-finished to a final thickness of 0.225 in.
The
edge contour was cut using electric-discharge machining forming an almost
T-shaped configuration. And then, the pole
tip was machined more precisely based on the coordinates that were
determined from a prior magnetic analysis.
A quarter iron core
consisted of 18 laminations layered together and separated by a 0.005-in kapton insulation. They were bound together by 2 bolts and
a pin through the
layer; the pin was spot-welded at both ends
to the iron.
The
quarter sections were assembled together to form the full magnet iron core
using a tight-tolerance alignment
jig, and then bolted and pinned together. It should be noted that a tongue-and-groove
edge was machined into each quarter core in order to allow self-fixturing during final assembly.
(See Fig. xx for the full core assembly).
In
order to install the coils into the iron core later, however, the pins were removed
and the quarter sections were disassembled.
4.2 Coil
The
hollow coil conductor was an oxygen-free high conductivity (OFHC) copper alloy,
8 mm. square with a 4.3 mm. inside diameter for water passage. Each coil pack
was composed of 4 layers of conductor connected in series electrically. But in
order to minimize the pressure drop and the increase in cooling water
temperature, they were split into 2 water passages. Each layer consisted of 13
turns with a 0.020-in kapton insulation wrapped
around the conductor in a half-lap pattern.
The
coil pack was epoxy-impregnated in a special mold. The procedure called for a reasonable
tolerance specification in order to enable us to assemble the pack into the
iron pole. Figure 5 shows the full coil pack.
Figure 5: Coil Pack 1
C:
Final Assembly
Each quarter section of the iron core was
fitted with a coil pack, and then they were assembled together into a full
magnet. Then all the other copper busses and other hardware were installed and
the whole magnet was mounted into its support structure for testing. Prior to
installation into the AGS ring, however, each magnet was split into halves
first so that it can be installed around the existing beam pipe. Figure 6 shows
the final assembly of the whole magnet.
Figure 6: Full Assembly During
Testing
5 Measurements
The
magnetic multipoles at a given radius (r) and distance z from
the center of the quadrupole
are expressed as Br=Sbn(r,z)·cos[(n+1)q]{n=1
Quad…} The integrated strengths Bn=∫bn(r,z)·dz of the
magnetic multipoles of the “thin” quadrupoles
were measured with the rotating coil method and also calculated using the
results from the 3D simulations. The Integrated quadrupole
strength of the “thin” quad at r=7 [cm] I=310 [A] and the ratios Rn=104·∫bn(r,z)dz/∫b1(r,z)dz of the allowed multipoles appear in Table 2.
Table 2. The integrated B1 strength
and the ratios Rn of the first three
allowed multipoles at r=7 [cm].
|
T [GeV] |
B1[T] |
R5 |
R9 |
R13 |
|
Calc |
0.88 |
-20 |
-210 |
-35 |
|
Meas |
0.89 |
+55 |
-45 |
-13 |
6 CONCLUSIONS
We designed and built a “thin” quadrupole,
10 cm long. The measured strength of the first allowed multipole
(dodecapole) was below the specified value. The
lamination thickness which was determined with transient field calculations
generated the expected magnetic multipoles and ohmic losses in the
laminations.
Figure
7: An isometric view of the current density (Jeddy)
at a particular cross section of the laminations, and time. The cross section of the
lamination is shown by the rectangle.
Figure 8: Same as in Fig. 5 but at a later time.
Note the overlapping of the Jeddy. This
overlapping causes the cancellation of the eddy currents, thus reduction of the
ohmic losses due to eddy currents.
6
References
[1] H. Huang, et al.,
Proc. EPAC06, (2006), p. 273.
[2] OPERA computer
code. Vector Fields Inc.